Vol. 11, No. 1, 2017

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Logarithmic good reduction, monodromy and the rational volume

Arne Smeets

Vol. 11 (2017), No. 1, 213–233
Abstract

Let R be a strictly local ring complete for a discrete valuation, with fraction field K and residue field of characteristic p > 0. Let X be a smooth, proper variety over K. Nicaise conjectured that the rational volume of X is equal to the trace of the tame monodromy operator on -adic cohomology if X is cohomologically tame. He proved this equality if X is a curve. We study his conjecture from the point of view of logarithmic geometry, and prove it for a class of varieties in any dimension: those having logarithmic good reduction.

Keywords
étale cohomology, logarithmic geometry, monodromy, nearby cycles, rational points
Mathematical Subject Classification 2010
Primary: 14F20
Secondary: 11G25, 11S15
Milestones
Received: 17 March 2016
Revised: 6 July 2016
Accepted: 10 August 2016
Published: 23 January 2017
Authors
Arne Smeets
Radboud Universiteit Nijmegen
IMAPP
Heyendaalseweg 135
6525 AJ Nijmegen
Netherlands University of Leuven
Departement Wiskunde
Celestijnenlaan 200B
3001 Heverlee
Belgium